1 / 20

Green’s function of a dressed particle (today: Holstein polaron)

Green’s function of a dressed particle (today: Holstein polaron). Mona Berciu, UBC. Collaborators: Glen Goodvin, George Sawaztky, Alexandru Macridin More details: M. B., PRL 97, 036402 (2006) G. G., M. B. and G. S., to be posted soon on the archives

chutchison
Télécharger la présentation

Green’s function of a dressed particle (today: Holstein polaron)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Green’s function of a dressed particle (today: Holstein polaron) Mona Berciu, UBC Collaborators: Glen Goodvin,George Sawaztky,Alexandru Macridin More details: M. B., PRL 97, 036402 (2006) G. G., M. B. and G. S., to be posted soon on the archives Funding: NSERC, CIAR Nanoelectronics, Sloan Foundation

  2. Motivation: Old problem: try to understand properties of a dressed particle, e.g. electron dressed by phonons (polaron), or spin-waves, or orbitronic deg. of freedom, or combinations of these and other bosonic excitations. For a single particle, the quantity of most interest is its Green’s function: poles give us the whole one-particle spectrum, residues have partial information about the eigenstates. Note: there is a substantial amount of work dedicated to finding only low-energy (GS) properties. We want the full Green’s function; we want a simple yet accurate approximation that works decently for all values of the coupling strength, so that we can understand regimes where perturbation does not work!

  3. Holstein Hamiltonian– describes one of the simplest (on-site, linear) electron-phonon couplings

  4. weak coupling Lang-Firsov impurity limit E k W How does the spectral weight evolve between these two very different limits?

  5. Calculate the Green’s function: use Dyson’s identity repeatedly, generate infinite hierarchy We can solve these exactly if t=0 or g=0. For finite t and g, make Momentum Average approximation: Note: this is exact if t=0 (no k dependence)  MA should work well at least for strong coupling g/t>>1, where there was no good approximation for G (perturbation theory gives only the GS, not the whole G)

  6. The MA end result:: exact for both t=0, g=0 Other aproximations: (a) simple to evaluate: (b) numerically intensive  Diagrammatic Quantum Monte Carlo (QMC) – in principle exact summation of all diagrams  exact diagonalizations (various cutoffs for Hilbert space),

  7. Comparisons: (I) GS results in 1D

  8. Agreement becomes better with increasing d, but MA calculations just as easy (fractions of second)

  9. 3D: no QMC results, but data is very persuasive

  10. How about higher-energy results? (much fewer “exact” numerical results). numerics: G. De Filippis et al., PRB 72, 014307 (2005)

  11. Diagrammatic meaning of MA : sums ALL self-energy diagrams, but each free propagator is momentum averaged, i.e. any is replaced by Example: 1nd order diagram: Higher order MA diagrams can be similarly grouped together and summed exactly.

  12. Why is summing ALL diagrams (even if with approx. expressions) BETTER than summing only some (exact) diagrams: understanding thesum rules of the spectral weight • Comments: • Knowing all the sum rules  exact Green’s function • Sum rules can be calculated exactly, with enough patience. • Traditional method of computation – using equations of motion [P.E. Kornilovitch, EPL 59, 735 (2002)] • (ii) Sum rules must be the same irrespective of strength of coupling. • Mn has units of energyn combinations of the various energy scales at the right power. • (iv) Traditional wisdom: the more sum rules are obeyed, the better the approximation is. WRONG!

  13. How to calculate the sum rules for the MA approximation?

  14. Alternative way to compute sum rules: use perturbational expansion (diagrammatics) for small coupling + + + + + + + + + + + + + + + MA vs SCBA: -0th order diagram correct  M0 and M1 are exact for both approximations; -1st order diagram correct  M2 and M3 are exact for both approximations; -2nd order diagrams: SCBA misses 1  M4 missing one g4 term (important at large g) MA still exact for M4, M5. Errors appear in M6, but not in the dominant terms.

  15. Conclusions: If t>>g  dominant term is . Both approximations get it correctly  ok behavior If g>>t  dominant term is ~ or -- MA gets is exactly (after all, it is exact for t=0)  ok behavior. On the other hand, SCBA does really poorly for large g. Keeping all diagrams, even if none is exact, may be more important than summing exactly a subclass of diagrams. On to more results ….

  16. Momentum-dependent low-energy results:

  17. W W W l l l

  18. The answer to my initial question, according to the Momentum Average (MA) approximation: l l l l

More Related